What a proportion is
Two quantities are proportional when they change by the same factor. That lets you answer questions like “if this costs that, how much does this other amount cost?” with three known numbers. The method is known as the rule of three, and in class you probably met it as solving for x by cross-multiplying.
Direct proportion
Both quantities grow together. If 3 pounds of apples cost $12, 5 pounds cost:
x = 12 × 5 ÷ 3 = 20
The general form: if a corresponds to b, then c corresponds to x = b × c ÷ a.
Scaling a recipe is the classic use. If 2 cups of flour make 12 cookies, 30 cookies need 2 × 30 ÷ 12 = 5 cups.
Inverse proportion
When one quantity goes up, the other goes down by the same factor. If 4 workers finish a job in 6 days, 8 workers finish it in:
x = 4 × 6 ÷ 8 = 3
The general form flips: x = a × b ÷ c. Driving works the same way: a trip that takes 3 hours at 60 mph takes 4 hours at 45 mph.
How to tell which one you need
Ask yourself: if I double the first quantity, what happens to the second?
- It doubles too: direct proportion.
- It halves: inverse proportion.
- Something else: neither, and the rule of three doesn’t apply.
Percentages are proportions too
15% of $240 is the proportion “100 is to 15 as 240 is to x”, so x = 15 × 240 ÷ 100 = 36. For everyday percentage questions like discounts and sales tax, the percentage calculator is quicker.
When it doesn’t work
The rule of three assumes a straight-line relationship, and plenty of real ones aren’t:
- Baking times don’t double when you double a recipe.
- Bulk pricing: the bigger pack is usually cheaper per pound.
- Fuel use rises much faster than speed on the highway.
- Teamwork: eight people can’t always work four times as fast as two, because they get in each other’s way.
When in doubt, check the answer against common sense. If the math says a job can be done in ten minutes by a hundred people, it probably can’t.